volume of solid of revolution

Let us consider a solid of revolution, which is generated when a planar domain D rotates about a line of the same plane. We chose this line for the x-axis, and for simplicity we assume that the boundaries of D are the mentioned axis, two ordinatesx=a,  x=b(>a), and a continuousMathworldPlanetmath curve   y=f(x).

Between the bounds a anb b we fit a sequence of points  x1,x2,,xn-1  and draw through these the ordinates which divide the domain D in n parts. Moreover we form for every part the (maximal) inscribedMathworldPlanetmath and the (minimal) circumscribedMathworldPlanetmath rectangleMathworldPlanetmath. In the revolution of D, each rectangle generates a circular cylinder. The considered solid of revolution is part of the volume V> of the union of the cyliders generated by the circumscribed rectangles and at the same time contains the volume V< of the union of the cylinders generated by the inscribed rectangles.

Now it is apparent that


where  M1,M2,,Mn  are the greatest and  m1,m2,,mn  the least values of the continuous function f on the intervalsMathworldPlanetmathPlanetmath (http://planetmath.org/Interval)   [a,x1],  [x1,x2], …, [xn-1,b]. The volume V of the solid of revolution thus satisfies


and this is true for any   x1<x2<<xn-1  of the interval  [a,b]. The theory of the Riemann integral guarantees that there exists only one real number V having this property and that it is also the definition of the integral abπ[f(x)]2𝑑x. Therefore the volume of the given solid of revolution can be obtained from



  • 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos.  Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
Title volume of solid of revolution
Canonical name VolumeOfSolidOfRevolution
Date of creation 2013-03-22 17:20:12
Last modified on 2013-03-22 17:20:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Topic
Classification msc 51M25
Related topic PappussTheoremForSurfacesOfRevolution
Related topic SurfaceOfRevolution
Related topic VolumeAsIntegral